Superstripes and the excitation spectrum of a spin-orbit-coupled...

ISTITUTO NAZIONALE

DI OTTICA

INO

UNIVERSITA DEGLI STUDI

DI TRENTO

Superstripes and the excitation spectrum of

a spin-orbit-coupled BEC

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii,and Sandro Stringari

Trieste, 2013 May

Breaking of two symmetries

Bose condensation

of defectons

A. F. Andreev and I. M. Lifshitz

JETP 29, 1107 (1969)

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Breaking of two symmetries

Bose condensation

of defectons

A. F. Andreev and I. M. Lifshitz

JETP 29, 1107 (1969)

Soft-core interactions

0 0.5 1 1.5 2

x

0

0.5

1

1.5

y

0 0.5 1 1.5 2

x

0

0.5

1

1.5

y

(a)

(c) (d)

(b)

N. Henkel, et al., PRL 104, 195302 (2010)

F. Cinti, et al., PRL 105, 135301 (2010)


Spin-orbit coupled BEC (single particle picture)

bulk, no interactions, Ω = 0

−2 −1 0 1 2

−0.5

0

0.5

1

p / k0

E(p

) / k

02 |↓⟩ |↑⟩

BEC

Single-particle Hamiltonian

h0 =1

2

ˆ

(px − k0σz)2 + p2

⊥

˜

+Ω

2σx +

δ

2σz

Equal Rashba and Dresselhauscouplings

⇐= U−1 hlab0 U

U = eiΘ(x)σz/2



bulk, no interactions, Ω 6= 0

−2 −1 0 1 2

−0.5

0

0.5

1

p / k0

E(p

) / k

02 |↓⟩ |↑⟩|↓⟩|↑⟩

BEC


h0 =1

2

ˆ

(px − k0σz)2 + p2

⊥

˜

+Ω

2σx +

δ

2σz


⇐= U−1 hlab0 U

U = eiΘ(x)σz/2

±k1 = ±k0

s

1 − Ω2

4k40



bulk, no interactions, Ω 6= 0

−2 −1 0 1 2

−0.5

0

0.5

1

p / k0

E(p

) / k

02 |↓⟩ |↑⟩|↓⟩|↑⟩

BEC


h0 =1

2

ˆ

(px − k0σz)2 + p2

⊥

˜

+Ω

2σx +

δ

2σz


7

6

5

4

3

2

1

0

gnil

pu

oC

na

ma

R/EL

-1.0 -0.5 0.0 0.5 1.0

Minima location in units of kL

±k1 = ±k0

s

1 − Ω2

4k40

Lin et al, Nature

83, 1523 (2010)


Many-body ground state

Three quantum phases γ = (g − g↑↓) / (g + g↑↓) > 0 , n(c) = k20/ (2γg)

(I). k1 6= 0, C+ = C−,〈σz〉 = 0

(II). k1 6= 0, C− = 0 orC+ = 0, 〈σz〉 6= 0

(III). k1 = 0, 〈σz〉 = 0

ψ =

„

ψ↑

ψ↓

«

=√n

»

C+

„

cos θ− sin θ

«

eik1x +C−

„

sin θ− cos θ

«

e−ik1x

–

cos 2θ =k1

k0, 〈σz〉 =

k1

k0

`

|C+|2 − |C−|2´

LY, Pitaevskii, Stringari, PRL 108, 225301 (2012)




(I). k1 6= 0, C+ = C−,〈σz〉 = 0

(II). k1 6= 0, C− = 0 orC+ = 0, 〈σz〉 6= 0

(III). k1 = 0, 〈σz〉 = 0

Ω(I-II)cr = 2k2

0

p

2γ/(1 + 2γ) small for 87Rb

Ω(II-III)cr = 2k2

0

1.0

0.5

0.03210

Hold time th (s)

= 0.14(3) s

Dynamics at = 0.6 EL

1.0

0.8

0.6

0.4

0.2

0.0

,n

oitar

ap

es

es

ah

Ps

0.012 3 4 5 6 7

0.12 3 4 5 6 7

12

Raman Coupling /EL

C = 0.20(2) EL

Ω

Ω

τ

Ω




−2 −1 0 1 20.6

0.8

1

1.2

1.4x 10

13

k1x / π

| ψ↑,

↓ (

x) |2 [

cm−

3 ]

Ω(I-II)cr = 2k2

0

p

2γ/(1 + 2γ) small for 87Rb

Ω(II-III)cr = 2k2

0

To increase the effect of the contrast, choose larger values of γ


Excitation spectrum in phase II

−2 −1.5 −1 −0.5 0 0.51

2

3

ω+ /

k 02

−2 −1.5 −1 −0.5 0 0.50

0.1

0.2

0.3

q / k0

ω− /

k 02

G1/k20 = 0.5, G2/k

20 = 0.12

Ω/k20 = 1.22, 1.33, 1.46

Despite spinor nature, occurrenceof Raman coupling gives rise to asingle gapless branch

Emergence of a roton minimumat finite q: a tendency of thesystem towards crystallization

Martone, LY, Pitaevskii and Stringari, PRA 86, 063621(2012)


Excitation spectrum in phase II

−2 −1 0 1 20

0.2

0.4

E(p

)

−2 −1.5 −1 −0.5 0 0.50

0.1

0.2

0.3

q / k0

ω− /

k 02

G1/k20 = 0.5, G2/k

20 = 0.12

Ω/k20 = 1.22, 1.33, 1.46

Despite spinor nature, occurrenceof Raman coupling gives rise to asingle gapless branch

Emergence of a roton minimumat finite q: a tendency of thesystem towards crystallization

Martone, LY, Pitaevskii and Stringari, PRA 86, 063621(2012)


Excitation spectrum in phase I

Equilibrium state

−2 −1 0 1 20.6

0.8

1

1.2

1.4x 10

13

k1x / π

| ψ↑,

↓ (

x) |2 [

cm−

3 ]

G1/k20 = 0.3, G2/k2

0 = 0.08, Ω/k20 = 1

Translational invariance symmetry breaking

U(1) symmetry breaking

„

ψ0↑

ψ0↓

«

=P

K

„

a−k1+K

−b−k1+K

«

ei(K−k1)x, K is the reciprocal lattice vector



Bogoliubov + Bloch theory

„

ψ↑

ψ↓

«

= e−iµt

»„

ψ0↑

ψ0↓

«

+

„

u↑(r)u↓(r)

«

e−iωt +

„

v∗↑(r)v∗↓(r)

«

eiωt

–

uq ↑, ↓(r) = e−ik1xX

K

Uq ↑, ↓ K eiq·r+iKx

vq ↑, ↓(r) = eik1xX

K

Vq ↑, ↓ K eiq·r−iKx

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

Emergence of two gapless bands

Vanishing of the frequency atthe edge of the Brillouin zone

Divergent behavior of staticstructure factor in density channel



Superfluid Supersolid

a)

0 0.2 0.4 0.6 0.8 1

ka/2π

0

5

10

15

20

ω

a)a)a)a) b)

0 0.2 0.4 0.6 0.8

ka/2π

0

5

10

15

20

25

ω

b)b)b)

Soft-coreinteractions

S. Saccani et al., PRL 108, 175301 (2012)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

Emergence of two gapless bands

Vanishing of the frequency atthe edge of the Brillouin zone

Divergent behavior of staticstructure factor in density channel


Density structure factor

0

0.5

1

1.5

2 0

0.4

0.8

1.2

1.60

0.5

1

ω / k02

qx / k

1

S(ω

, qx)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

0 0.5 1 1.5 20

0.5

1

qx / k

1

S(q

x)



0

0.5

1

1.5

2 0

0.4

0.8

1.2

1.60

0.5

1

ω / k02

qx / k

1

S(ω

, qx)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

0 0.5 1 1.5 20

0.5

1

qx / k

1

S(q

x)Upper branch isa density waveat small qx



0

0.5

1

1.5

2 0

0.4

0.8

1.2

1.60

0.5

1

ω / k02

qx / k

1

S(ω

, qx)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

0 0.5 1 1.5 20

0.5

1

qx / k

1

S(q

x)Upper branch isa density waveat small qx

S(qx) for thelower branchdiverges


Spin structure factor

0

0.5

1

1.5

2 0

0.4

0.8

1.2

1.60

0.5

1

ω / k02

qx / k

1

Sσ(ω

, qx)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

0 0.5 1 1.5 20

0.5

1

qx / k

1

Sσ(q

x)


Spin structure factor

0

0.5

1

1.5

2 0

0.4

0.8

1.2

1.60

0.5

1

ω / k02

qx / k

1

Sσ(ω

, qx)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

0 0.5 1 1.5 20

0.5

1

qx / k

1

Sσ(q

x)Lower branch isa spin waveat small qx


Sum rule approach

Define qx-component density operator F and (qx − qB)-componentmomentum density operator G

F =X

j

eiqxxj , 〈 [F , G] 〉 = qxN〈eiqBx〉

G =X

j

h

pxj e−i(qx−qB)xj + e−i(qx−qB)xjpxj

i

/2

p−th moments:

mp(O) =P

ℓ(|〈0|O|ℓ〉|2

+|〈0|O†|ℓ〉|2)ωpℓ0

qx → qB F G

m1 q2x (qx − qB)2

m0 S(qx) |qx − qB |m−1 χ(qx)

Bogoliubov inequality: m−1(F )m1(G) ≥ |〈 [F ,G] 〉|2 χ(qx) ∝ 1/(qx − qB)2

Uncertainty inequality: m0(F )m0(G) ≥ |〈 [F ,G] 〉|2 S(qx) ∝ 1/|qx − qB |


Sound velocity

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

Ω / k02

c / k

0

Upper branch

Lower branch

I−II phasetransition

c(d, s)⊥

> c(d, s)x , inertia of flow caused by stripes

c(d)⊥

≃

√

(g + g↑↓)n/2, well reproduced by usual Bogoliubov soundvelocity


Conclusions

Excitation spectrum in the stripephase: double gapless band structure

At small wave vector the lower andupper branches have, respectively, aspin and density nature

Close to the first Brillouin zone thelower branch acquires an importantdensity character, S(qx) diverges

ω /

k 02

qx / k

1

0 0.5 1 1.5 20

0.5

1

1.5

ω /

k 02

qx / k

1

0 0.5 1 1.5 20

0.5

1

1.5

LY, Martone, Pitaevskii, Stringari

arXiv: 1303.6903


Thank you !

Superstripes and the excitation spectrum of a spin-orbit-coupled...

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